Abstract
In this paper we carry out a systematic study of a natural covering problem, used for identification across several areas, in the realm of parameterized complexity. In the Test Cover problem we are given a set [n] = {1,…,n} of items together with a collection, \(\cal T\), of distinct subsets of these items called tests. We assume that \(\cal T\) is a test cover, i.e., for each pair of items there is a test in \(\cal T\) containing exactly one of these items. The objective is to find a minimum size subcollection of \(\cal T\), which is still a test cover. The generic parameterized version of Test Cover is denoted by \(p(k,n,|{\cal T}|)\)-Test Cover. Here, we are given \(([n],\cal{T})\) and a positive integer parameter k as input and the objective is to decide whether there is a test cover of size at most \(p(k,n,|{\cal T}|)\). We study four parameterizations for Test Cover and obtain the following:
(a) k-Test Cover, and (n − k)-Test Cover are fixed-parameter tractable (FPT), i.e., these problems can be solved by algorithms of runtime \(f(k)\cdot poly(n,|{\cal T}|)\), where f(k) is a function of k only.
(b) \((|{\cal T}|-k)\)-Test Cover and (logn + k)-Test Cover are W[1]-hard. Thus, it is unlikely that these problems are FPT.
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Crowston, R., Gutin, G., Jones, M., Saurabh, S., Yeo, A. (2012). Parameterized Study of the Test Cover Problem. In: Rovan, B., Sassone, V., Widmayer, P. (eds) Mathematical Foundations of Computer Science 2012. MFCS 2012. Lecture Notes in Computer Science, vol 7464. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32589-2_27
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DOI: https://doi.org/10.1007/978-3-642-32589-2_27
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